metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C12).1Q8, C6.12(C4⋊Q8), C2.7(C12⋊Q8), (C2×C4).8Dic6, C6.2(C4⋊D4), (C2×Dic3).1Q8, (C2×Dic3).8D4, (C22×C4).84D6, C2.8(Dic3⋊D4), C22.39(S3×Q8), C2.8(D6⋊Q8), C22.151(S3×D4), C6.21(C22⋊Q8), C2.8(C23.9D6), C2.8(Dic3.Q8), C6.C42.5C2, (C22×C12).7C22, C6.11(C42.C2), C2.4(C12.6Q8), C22.41(C2×Dic6), C22.84(C4○D12), C2.C42.14S3, C23.363(C22×S3), (C22×C6).282C23, C6.4(C22.D4), C22.82(D4⋊2S3), C3⋊1(C23.81C23), C2.9(Dic3.D4), (C22×Dic3).7C22, (C2×C6).63(C2×Q8), (C2×C6).194(C2×D4), (C2×C4⋊Dic3).7C2, (C2×C6).57(C4○D4), (C2×Dic3⋊C4).6C2, (C3×C2.C42).9C2, SmallGroup(192,216)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.(C4⋊Q8)
G = < a,b,c,d | a2=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=ab7, dbd-1=ab5, dcd-1=ac-1 >
Subgroups: 368 in 150 conjugacy classes, 61 normal (51 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C4⋊C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C4⋊C4, Dic3⋊C4, C4⋊Dic3, C22×Dic3, C22×C12, C23.81C23, C6.C42, C3×C2.C42, C2×Dic3⋊C4, C2×C4⋊Dic3, C6.(C4⋊Q8)
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C22×S3, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C2×Dic6, C4○D12, S3×D4, D4⋊2S3, S3×Q8, C23.81C23, C12.6Q8, Dic3.D4, C23.9D6, Dic3⋊D4, C12⋊Q8, Dic3.Q8, D6⋊Q8, C6.(C4⋊Q8)
►Regular action on 192 points
Generators in S192
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 49)(11 50)(12 51)(13 181)(14 182)(15 183)(16 184)(17 185)(18 186)(19 187)(20 188)(21 189)(22 190)(23 191)(24 192)(25 129)(26 130)(27 131)(28 132)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 146)(38 147)(39 148)(40 149)(41 150)(42 151)(43 152)(44 153)(45 154)(46 155)(47 156)(48 145)(61 136)(62 137)(63 138)(64 139)(65 140)(66 141)(67 142)(68 143)(69 144)(70 133)(71 134)(72 135)(73 104)(74 105)(75 106)(76 107)(77 108)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 165)(86 166)(87 167)(88 168)(89 157)(90 158)(91 159)(92 160)(93 161)(94 162)(95 163)(96 164)(109 179)(110 180)(111 169)(112 170)(113 171)(114 172)(115 173)(116 174)(117 175)(118 176)(119 177)(120 178)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)
(1 40 184 105)(2 156 185 81)(3 42 186 107)(4 146 187 83)(5 44 188 97)(6 148 189 73)(7 46 190 99)(8 150 191 75)(9 48 192 101)(10 152 181 77)(11 38 182 103)(12 154 183 79)(13 108 49 43)(14 84 50 147)(15 98 51 45)(16 74 52 149)(17 100 53 47)(18 76 54 151)(19 102 55 37)(20 78 56 153)(21 104 57 39)(22 80 58 155)(23 106 59 41)(24 82 60 145)(25 176 133 158)(26 113 134 85)(27 178 135 160)(28 115 136 87)(29 180 137 162)(30 117 138 89)(31 170 139 164)(32 119 140 91)(33 172 141 166)(34 109 142 93)(35 174 143 168)(36 111 144 95)(61 167 132 173)(62 94 121 110)(63 157 122 175)(64 96 123 112)(65 159 124 177)(66 86 125 114)(67 161 126 179)(68 88 127 116)(69 163 128 169)(70 90 129 118)(71 165 130 171)(72 92 131 120)
(1 140 184 32)(2 70 185 129)(3 138 186 30)(4 68 187 127)(5 136 188 28)(6 66 189 125)(7 134 190 26)(8 64 191 123)(9 144 192 36)(10 62 181 121)(11 142 182 34)(12 72 183 131)(13 29 49 137)(14 126 50 67)(15 27 51 135)(16 124 52 65)(17 25 53 133)(18 122 54 63)(19 35 55 143)(20 132 56 61)(21 33 57 141)(22 130 58 71)(23 31 59 139)(24 128 60 69)(37 116 102 88)(38 179 103 161)(39 114 104 86)(40 177 105 159)(41 112 106 96)(42 175 107 157)(43 110 108 94)(44 173 97 167)(45 120 98 92)(46 171 99 165)(47 118 100 90)(48 169 101 163)(73 166 148 172)(74 91 149 119)(75 164 150 170)(76 89 151 117)(77 162 152 180)(78 87 153 115)(79 160 154 178)(80 85 155 113)(81 158 156 176)(82 95 145 111)(83 168 146 174)(84 93 147 109)
G:=sub<Sym(192)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,49)(11,50)(12,51)(13,181)(14,182)(15,183)(16,184)(17,185)(18,186)(19,187)(20,188)(21,189)(22,190)(23,191)(24,192)(25,129)(26,130)(27,131)(28,132)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,146)(38,147)(39,148)(40,149)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,145)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,133)(71,134)(72,135)(73,104)(74,105)(75,106)(76,107)(77,108)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,165)(86,166)(87,167)(88,168)(89,157)(90,158)(91,159)(92,160)(93,161)(94,162)(95,163)(96,164)(109,179)(110,180)(111,169)(112,170)(113,171)(114,172)(115,173)(116,174)(117,175)(118,176)(119,177)(120,178), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,40,184,105)(2,156,185,81)(3,42,186,107)(4,146,187,83)(5,44,188,97)(6,148,189,73)(7,46,190,99)(8,150,191,75)(9,48,192,101)(10,152,181,77)(11,38,182,103)(12,154,183,79)(13,108,49,43)(14,84,50,147)(15,98,51,45)(16,74,52,149)(17,100,53,47)(18,76,54,151)(19,102,55,37)(20,78,56,153)(21,104,57,39)(22,80,58,155)(23,106,59,41)(24,82,60,145)(25,176,133,158)(26,113,134,85)(27,178,135,160)(28,115,136,87)(29,180,137,162)(30,117,138,89)(31,170,139,164)(32,119,140,91)(33,172,141,166)(34,109,142,93)(35,174,143,168)(36,111,144,95)(61,167,132,173)(62,94,121,110)(63,157,122,175)(64,96,123,112)(65,159,124,177)(66,86,125,114)(67,161,126,179)(68,88,127,116)(69,163,128,169)(70,90,129,118)(71,165,130,171)(72,92,131,120), (1,140,184,32)(2,70,185,129)(3,138,186,30)(4,68,187,127)(5,136,188,28)(6,66,189,125)(7,134,190,26)(8,64,191,123)(9,144,192,36)(10,62,181,121)(11,142,182,34)(12,72,183,131)(13,29,49,137)(14,126,50,67)(15,27,51,135)(16,124,52,65)(17,25,53,133)(18,122,54,63)(19,35,55,143)(20,132,56,61)(21,33,57,141)(22,130,58,71)(23,31,59,139)(24,128,60,69)(37,116,102,88)(38,179,103,161)(39,114,104,86)(40,177,105,159)(41,112,106,96)(42,175,107,157)(43,110,108,94)(44,173,97,167)(45,120,98,92)(46,171,99,165)(47,118,100,90)(48,169,101,163)(73,166,148,172)(74,91,149,119)(75,164,150,170)(76,89,151,117)(77,162,152,180)(78,87,153,115)(79,160,154,178)(80,85,155,113)(81,158,156,176)(82,95,145,111)(83,168,146,174)(84,93,147,109)>;
G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,49)(11,50)(12,51)(13,181)(14,182)(15,183)(16,184)(17,185)(18,186)(19,187)(20,188)(21,189)(22,190)(23,191)(24,192)(25,129)(26,130)(27,131)(28,132)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,146)(38,147)(39,148)(40,149)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,145)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,133)(71,134)(72,135)(73,104)(74,105)(75,106)(76,107)(77,108)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,165)(86,166)(87,167)(88,168)(89,157)(90,158)(91,159)(92,160)(93,161)(94,162)(95,163)(96,164)(109,179)(110,180)(111,169)(112,170)(113,171)(114,172)(115,173)(116,174)(117,175)(118,176)(119,177)(120,178), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,40,184,105)(2,156,185,81)(3,42,186,107)(4,146,187,83)(5,44,188,97)(6,148,189,73)(7,46,190,99)(8,150,191,75)(9,48,192,101)(10,152,181,77)(11,38,182,103)(12,154,183,79)(13,108,49,43)(14,84,50,147)(15,98,51,45)(16,74,52,149)(17,100,53,47)(18,76,54,151)(19,102,55,37)(20,78,56,153)(21,104,57,39)(22,80,58,155)(23,106,59,41)(24,82,60,145)(25,176,133,158)(26,113,134,85)(27,178,135,160)(28,115,136,87)(29,180,137,162)(30,117,138,89)(31,170,139,164)(32,119,140,91)(33,172,141,166)(34,109,142,93)(35,174,143,168)(36,111,144,95)(61,167,132,173)(62,94,121,110)(63,157,122,175)(64,96,123,112)(65,159,124,177)(66,86,125,114)(67,161,126,179)(68,88,127,116)(69,163,128,169)(70,90,129,118)(71,165,130,171)(72,92,131,120), (1,140,184,32)(2,70,185,129)(3,138,186,30)(4,68,187,127)(5,136,188,28)(6,66,189,125)(7,134,190,26)(8,64,191,123)(9,144,192,36)(10,62,181,121)(11,142,182,34)(12,72,183,131)(13,29,49,137)(14,126,50,67)(15,27,51,135)(16,124,52,65)(17,25,53,133)(18,122,54,63)(19,35,55,143)(20,132,56,61)(21,33,57,141)(22,130,58,71)(23,31,59,139)(24,128,60,69)(37,116,102,88)(38,179,103,161)(39,114,104,86)(40,177,105,159)(41,112,106,96)(42,175,107,157)(43,110,108,94)(44,173,97,167)(45,120,98,92)(46,171,99,165)(47,118,100,90)(48,169,101,163)(73,166,148,172)(74,91,149,119)(75,164,150,170)(76,89,151,117)(77,162,152,180)(78,87,153,115)(79,160,154,178)(80,85,155,113)(81,158,156,176)(82,95,145,111)(83,168,146,174)(84,93,147,109) );
G=PermutationGroup([[(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,49),(11,50),(12,51),(13,181),(14,182),(15,183),(16,184),(17,185),(18,186),(19,187),(20,188),(21,189),(22,190),(23,191),(24,192),(25,129),(26,130),(27,131),(28,132),(29,121),(30,122),(31,123),(32,124),(33,125),(34,126),(35,127),(36,128),(37,146),(38,147),(39,148),(40,149),(41,150),(42,151),(43,152),(44,153),(45,154),(46,155),(47,156),(48,145),(61,136),(62,137),(63,138),(64,139),(65,140),(66,141),(67,142),(68,143),(69,144),(70,133),(71,134),(72,135),(73,104),(74,105),(75,106),(76,107),(77,108),(78,97),(79,98),(80,99),(81,100),(82,101),(83,102),(84,103),(85,165),(86,166),(87,167),(88,168),(89,157),(90,158),(91,159),(92,160),(93,161),(94,162),(95,163),(96,164),(109,179),(110,180),(111,169),(112,170),(113,171),(114,172),(115,173),(116,174),(117,175),(118,176),(119,177),(120,178)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,40,184,105),(2,156,185,81),(3,42,186,107),(4,146,187,83),(5,44,188,97),(6,148,189,73),(7,46,190,99),(8,150,191,75),(9,48,192,101),(10,152,181,77),(11,38,182,103),(12,154,183,79),(13,108,49,43),(14,84,50,147),(15,98,51,45),(16,74,52,149),(17,100,53,47),(18,76,54,151),(19,102,55,37),(20,78,56,153),(21,104,57,39),(22,80,58,155),(23,106,59,41),(24,82,60,145),(25,176,133,158),(26,113,134,85),(27,178,135,160),(28,115,136,87),(29,180,137,162),(30,117,138,89),(31,170,139,164),(32,119,140,91),(33,172,141,166),(34,109,142,93),(35,174,143,168),(36,111,144,95),(61,167,132,173),(62,94,121,110),(63,157,122,175),(64,96,123,112),(65,159,124,177),(66,86,125,114),(67,161,126,179),(68,88,127,116),(69,163,128,169),(70,90,129,118),(71,165,130,171),(72,92,131,120)], [(1,140,184,32),(2,70,185,129),(3,138,186,30),(4,68,187,127),(5,136,188,28),(6,66,189,125),(7,134,190,26),(8,64,191,123),(9,144,192,36),(10,62,181,121),(11,142,182,34),(12,72,183,131),(13,29,49,137),(14,126,50,67),(15,27,51,135),(16,124,52,65),(17,25,53,133),(18,122,54,63),(19,35,55,143),(20,132,56,61),(21,33,57,141),(22,130,58,71),(23,31,59,139),(24,128,60,69),(37,116,102,88),(38,179,103,161),(39,114,104,86),(40,177,105,159),(41,112,106,96),(42,175,107,157),(43,110,108,94),(44,173,97,167),(45,120,98,92),(46,171,99,165),(47,118,100,90),(48,169,101,163),(73,166,148,172),(74,91,149,119),(75,164,150,170),(76,89,151,117),(77,162,152,180),(78,87,153,115),(79,160,154,178),(80,85,155,113),(81,158,156,176),(82,95,145,111),(83,168,146,174),(84,93,147,109)]])
42 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4F | 4G | ··· | 4N | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | - | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | Q8 | D6 | C4○D4 | Dic6 | C4○D12 | S3×D4 | D4⋊2S3 | S3×Q8 |
| kernel | C6.(C4⋊Q8) | C6.C42 | C3×C2.C42 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C2.C42 | C2×Dic3 | C2×Dic3 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 3 | 1 | 1 | 4 | 2 | 2 | 3 | 6 | 4 | 8 | 2 | 1 | 1 |
Matrix representation of C6.(C4⋊Q8) ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 8 |
| 0 | 0 | 0 | 0 | 11 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 2 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 6 |
| 0 | 0 | 0 | 0 | 3 | 10 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,11,11,0,0,0,0,8,2],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,10,6,0,0,0,0,7,3,0,0,0,0,0,0,1,0,0,0,0,0,11,12],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,11,4,0,0,0,0,2,2,0,0,0,0,0,0,3,3,0,0,0,0,6,10] >;
C6.(C4⋊Q8) in GAP, Magma, Sage, TeX
C_6.(C_4\rtimes Q_8)
% in TeX
G:=Group("C6.(C4:Q8)"); // GroupNames label
G:=SmallGroup(192,216);
// by ID
G=gap.SmallGroup(192,216);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,64,254,387,100,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^7,d*b*d^-1=a*b^5,d*c*d^-1=a*c^-1>;
// generators/relations